Question

Harry can rake the leaves in the yard 8 hours faster than his little brother Jimmy can. If they work together, they can complete the job in 3 hours. Using complete sentences, explain each step in figuring out how to determine the time it would take Jimmy to complete this job on his own.

Answer 1

The working together formula:


\[\frac{M \times N}{M + N} = t\]

M = time it takes one person to complete a job
N = time it takes another person to complete the same job
t = time it takes both to complete the job while working together

Answer 2

In this case let's represent the situation by the following:

\[\frac{H \times J}{H + J} = 3\]

H = hours it takes Harry to complete the job while working alone
J = hours it takes Jimmy to complete the job while working alone
3 = hours it takes both Harry and Jimmy to complete the job while working together.

Answer 3

@Hero wow thank you so much for the help, greatly appreciated!!!

Answer 4

Harry can do the job 8 hours faster than Jimmy, so the situation can be represented by
H = J - 8


So replacing H with J - 8 in the equation yields:

\[\frac{(J - 8) \times J}{(J - 8) + J} = 3\]

Perform the operations on the left side to get
\[\frac{J^2 - 8J}{2J - 8} = 3\]

Multiply both sides by 2J - 8 to get
\[J^2 - 8J = 3(2J - 8)\]

Distribute on the right side to get:
\[J^2 - 8J = 6J - 24\]

Move everything to the right side:
\[J^2 - 8J - 6J + 24 = 0\]

\[J^2 - 14J + 24 = 0\]

Now you have a factorable quadratic that remains. Factor the quadratic and continue solving for J.